Problem: Eudora ran from her home to her secret laboratory at an average speed of $12\text{ km/h}$. She then took one of her jetpacks and flew to her school at an average speed of $76\text{ km/h}$. Eudora traveled a total distance of $120$ kilometers, and the entire trip took $2$ hours. How long did Eudora spend running, and how long did she spend flying using her jetpack? Eudora ran for
Let $x$ represent the time (in hours) that Eudora spent running and let $y$ represent the time (in hours) that she spent flying. Since we have two unknowns, we need two equations to find them. Let's use the given information in order to write two equations containing $x$ and $y$. For instance, we are given that Eudora ran at an average speed of $\textit{12 km/h}$, flew an average speed of $\textit{76 km/h}$, and traveled a total distance of $\textit{120}$ kilometers. How can we model this sentence algebraically? The total distance Eudora ran can be modeled by $12x$, and the total distance she flew can be modeled by $76y$. Since these together add up to $120$, we get the following equation: $12x+76y=120$ We are also given that Eudora's entire trip took $\textit{2}$ hours. This can be expressed as: $x + y =2$ Now that we have a system of two equations, we can go ahead and solve it! We can now solve the system of equations by the elimination method. Note that the coefficient of $x$ in the first equation, $12$, is exactly $12$ times the coefficient of $x$ in the second equation, $1$. Therefore, we can multiply the second equation by ${-12}$ in order to eliminate $x$. $\begin{aligned} {-12}\cdot x+({-12})\cdot y&={-12}\cdot2\\\\ -12x-12y&=-24\end{aligned}$ Now we can eliminate $x$ : $\begin{aligned}12x+{76y}&=120\\\\ {+}\ -12x-{12y}&=-24\\ \hline\\ 0+64y &=96 \end{aligned}$ When we solve the resulting equation, we find that $y =1.5$, which we can substitute into $x+y=2$ to find that $x=0.5$. Recall that $x$ denotes the time that Eudora spent running and $y$ denotes the time she spent flying. Therefore, Eudora ran for $\textit{0.5}$ hours and flew for $\textit{1.5}$ hours using her jetpack.